Chapter 8 Scattering Theory - Particle Physics Group

CHAPTER 8. SCATTERING THEORY 140
where the factor (2l + 1) in the definition of the partial wave amplitudes f l (k) corresponds to
the degeneracy of the magnetic quantum number. (Some authors use different conventions,
like either dropping the factor (2l + 1) or including an additional factor 1/k in the definition
of f l .) The terms in the series (8.26) are known as a partial waves, which are simultaneous
eigenfunctions of the operators L 2 and L z with eigenvalues l(l + 1) 2 and 0, respectively. Our
aim is now to determine the amplitudes f l in terms of the radial functions R l (k,r) for solutions
(8.27) to the Schrödinger equation.
The radial equation. We recall the formula for the Laplacian in spherical coordinates
∆ = 1 (
∂
r 2 ∂ )
− L2
with − L r 2 ∂r ∂r 2 r 2 = 1 (
∂
sin θ ∂ )
+ 1 ∂ 2
2 sin θ ∂θ ∂θ sin 2 (8.28)
θ ∂ϕ 2
With the separation ansatz
u Elm (⃗x) = R El (r)Y lm (θ,ϕ) (8.29)
for the time-independent Schrödinger equation with central potential in spherical coordinates
[ (
{− 2 1 ∂
r 2 ∂ ) ] }
− L2
+ V (r) u(⃗x) = Eu(⃗x), (8.30)
2m r 2 ∂r ∂r 2 r 2
and L 2 Y lm (θ,ϕ) = l(l + 1) 2 Y lm (θ,ϕ) we find the radial equation
( (− 2 d
2
2m dr + 2 ) )
d l(l + 1)2
+ + V (r) R 2 r dr 2mr 2
El (r) = ER El (r). (8.31)
and its reduced form
( d
2
dr 2 + 2 r
)
d l(l + 1)
− − U(r) + k 2 R
dr r 2 l (k,r) = 0 (8.32)
with k = √ 2mE/ 2 and the reduced potential U(r) = (2m/ 2 )V (r).
Behavior near the center. For potentials less singular than r −2 at the origin the behavior
of R l (k,r) at r = 0 can be determined by expanding R l into a power series
∑ ∞
R l (k,r) = r s a n r n . (8.33)
Substitution into the radial equation (8.32) leads to the quadratic indicial equation with the
two solutions s = l and s = −(l + 1). Only the first one leads to a non-singular wave function
u(r,θ) at the origin r = 0.
Introducing a new radial function ˜R El (r) = rR El (r) and substituting into (8.31) leads to
the equation (
− 2
2m
n=0
)
d 2
dr + V eff(r) ˜R 2 El (r) = E ˜R El (r) (8.34)